HDMI carries video and audio. If you get an AV receiver, it will handle the audio and video. The speakers are attached to the receiver. The projector (or TV) is the hdmi output of the receiver. You connect your Xbox to one of the hdmi inputs of the receiver. You can attach a cable box, blu-ray player, etc. to the other hdmi inputs.
ETA. Those Logitech speakers will not work with a regular receiver. I'm talking about using regular speaker that you use wire for.